Question: The lifespans of sloths in a particular zoo are normally distributed. The average sloth lives $18.7$ years; the standard deviation is $4.3$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a sloth living between $5.8$ and $10.1$ years.
Explanation: $18.7$ $14.4$ $23$ $10.1$ $27.3$ $5.8$ $31.6$ $99.7\%$ $95\%$ $2.35\%$ $2.35\%$ We know the lifespans are normally distributed with an average lifespan of $18.7$ years. We know the standard deviation is $4.3$ years, so one standard deviation below the mean is $14.4$ years and one standard deviation above the mean is $23$ years. Two standard deviations below the mean is $10.1$ years and two standard deviations above the mean is $27.3$ years. Three standard deviations below the mean is $5.8$ years and three standard deviations above the mean is $31.6$ years. We are interested in the probability of a sloth living between $5.8$ and $10.1$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $99.7\%$ of the sloths will have lifespans within 3 standard deviations of the average lifespan. It also tells us that $95\%$ of the sloths will have lifespans within 2 standard deviations of the mean. That leaves $99.7\% - 95\% = 4.7\%$ of sloths between 2 and 3 standard deviations of the mean, or $2.35\%$ on either side of the distribution. The probability of a particular sloth living between $5.8$ and $10.1$ years is $\color{orange}{2.35\%}$.